Algebraic Connectivity for Vertex-Deleted Subgraphs, and a Notion of Vertex Centrality
Kirkland, Steve (2010) Algebraic Connectivity for Vertex-Deleted Subgraphs, and a Notion of Vertex Centrality. Discrete Mathematics, 310 (4). pp. 911-921. ISSN 0012-365X
Let G be a connected graph, suppose that v is a vertex of G, and denote the subgraph formed from G by deleting vertex v by G\v. Denote the algebraic connectivities of G and G\v by α(G) and (G\v), respectively. In this paper, we consider the functions ∅(v) = α(G)− α(G\v) and k(v) = α(G\v)/α(G), provide attainable upper and lower bounds on both functions, and characterise the equality cases in those bounds. The function yields a measure of vertex centrality, and we apply that measure to analyse certain graphs arising from food webs.
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